**Linear
Inequations **

We know that
in a statement in which two sides are equal is known as an equation. An
equation whose highest degree is one is known as linear equation.

For example,
2x + 3 = 5 and 5a – 7 = 0, etc.

A statement
in which two sides are related by the symbols >, <, ≤ or ≥ are called inequation. An inequation
whose highest degree is one is known as linear inequation.

For example,
3x – 2 < 12, 4y + 7 ≥ 15, etc.

In general,
a linear inequation can be written as

1. ax + b
< 0

2. ax + b
> 0

3. ax + b ≤ 0

4. ax + b ≥ 0

where a, b
are real numbers and a ≠ 0.

**Replacement
Set and Solution Set **

The set from
which the values of the variable in a linear inequation are chosen is called
the replacement set or universal set.

The set of
elements of the replacement set which satisfy the inequation when substituted
for the variable is called the solution set or truth set. The solution set depends
upon the replacement set.

For the
inequation x ≥ 5,

1. If replacement set is {3, 4, 5, 6, 7,
8, 9}, then the solution set is {5, 6, 7, 8, 9}.

2. If replacement set is {all natural
numbers}, then the solution set is {all natural numbers greater than or equal
to 5}

3. If replacement set is {–2, –1, 0, 1,
2, 3, 4}, then the solution set is ϕ.

**Properties
of Inequations **

1. Addition or subtraction of the same
integer from both sides of an inequation does not change the inequation, i.e.,
if a > b, then a + c > b + c and a – c > b – c.

2. Multiplying or dividing by the same
positive integer on both sides of an inequation does not change the inequation,
i.e., if a > b and c is any positive integer, then a × c > b × c and a/c >
b/c.

3. Multiplying or dividing by the same
negative integer on both sides of an inequation reverses the sign of inequality,
i.e., if a > b and c < 0, then a × c < b × c and a/c < b/c.

4. If the sides of an inequation are
interchanged, then the sign of inequality also reverses, i.e., if a > b,
then b < a.

**Example:**Solve 5x + 2 ≤ 22, x ϵ W.

**Solution:**Given inequation is, 5x + 2 ≤ 22

5x + 2 – 2 ≤ 22 – 2 (Subtracting
2 from both sides)

5x ≤ 20

x ≤ 4 (Dividing
both sides by 5)

Thus, the
solution set for 5x + 2 ≤ 22, x ϵ W is {0, 1, 2, 3, 4}.